1. Introduction
In closed basins, nighttime cooling frequently leads to the formation of pronounced inversions. The convergence of thermally driven downslope flows provides rapid initial cooling and the formation of a stable stratification at the basin floor. This stable stratification, combined with the confined geometry of the closed basin, which provides efficient sheltering with respect to the ambient flow, allows the in situ surface-driven cooling to remain concentrated in a shallow layer. Under favorable conditions, record minimum temperatures may occur at the floor of such basins. Temperature inversions of 10–15 K between the floor and the rim of the basin (over a depth on the order of 100 m) are common, and 30 K can be observed in more extreme cases.
In 2006, the Meteor Crater Experiment (METCRAX) field campaign was conducted in Arizona’s Meteor Crater (Whiteman et al. 2008). Measurements were designed to elucidate the mechanisms important in the buildup and destruction of the nighttime inversions. Unlike previously studied sinkhole basins (Clements et al. 2003; Whiteman et al. 2004; Steinacker et al. 2007), the Meteor Crater is to a good approximation rotationally symmetric and does not have any higher-elevation drainage areas. It was considered an ideal site for studying the undisturbed buildup of nighttime inversions. The temperature evolution, however, was found to be fundamentally different from that in previously studied basins of comparable size (Whiteman et al. 2008). While there was continuous cooling, a near-isothermal stratification persisted throughout the night in much of the crater atmosphere (Fig. 1). A surface-based strong stability layer was present only in the lowest 20% of the crater atmosphere. This behavior was observed during several nights and represented a recurring, systematic phenomenon. On the basis of the available data, a conceptual model was proposed (Whiteman et al. 2010). The conceptual model states that cold air that forms on the slightly sloping mesoscale plain outside the crater (Savage et al. 2008) interacts with the crater topography. While part of the flow splits around the crater, an upper portion is lifted up the crater rim, flows down the inner sidewall of the crater, and leads to a cooling pattern that is different from what would be expected from purely local cooling.
In this paper the conceptual picture suggested by Whiteman et al. (2010) is tested within the framework of a mass flux model, which is solved analytically and numerically. The central questions we would like to answer are (i) whether an intrusion of cold air from the outside can cause the basin atmosphere to become near-isothermal, and (ii) what mechanism causes the isothermalcy.
The problem addressed in this study is of interest also in the wider context of basin and valley meteorology. Decoupling of the surface layer from the flow aloft and the formation of cold-air pools occur frequently in complex terrain. The influence of synoptic or mesoscale flows on persistent cold-air pools in large basins and valleys has been investigated by Whiteman et al. (1999), Zhong et al. (2001), Zängl (2005a,b), and Reeves and Stensrud (2009), among others. These studies show the importance of cold-air advection, as compared to turbulent erosion, radiation, and other mechanisms, in the dissipation of cold-air pools. In small-scale basins, where more complete datasets are available from field studies, the process of mixing-induced warm intrusions and their effect on inversion development has been documented (Eisenbach et al. 2003). The peculiar phenomenon of mesoscale cold-air intrusions in the case of the Meteor Crater offers a small-scale analog to the cold-air advection in larger basins, which in this way can be investigated in more detail.
The paper is organized as follows: section 2 presents the mass flux model and some analytical solutions for the simplified case of intrusion-only cooling. The combined effect of in situ and intrusion cooling is investigated numerically in section 3. The physical significance of the near-isothermal stratification is discussed, and the effect of incomplete mixing of detrained air is studied.
2. Mass flux model
a. Model equations
To understand how cold-air intrusions affect the temperature evolution inside a closed basin, a mass flux and heat budget model is formulated. The hypothesis to be tested is that such intrusions can lead to a near-isothermal stratification above a strongly stratified surface layer. To reduce the problem to its essentials and make an analytical solution possible, we first consider a two-dimensional basin geometry with vertical sidewalls. In a further step we take into account the actual topography of the Meteor Crater.
Cold air entering the crater because of its negative buoyancy must to some extent detrain into the crater interior because of interfacial mixing, thereby cooling the crater. As a consequence of the detrainment, the vertical mass flux decreases downward. At some height the flow may reach its level of neutral buoyancy (LNB) h, where the remaining cold air detrains completely. It should be pointed out that the more general concept of buoyancy sorting and detrainment has been developed for mass flux parameterizations of moist convection (Emanuel 1991), where evaporatively cooled mixtures of cloudy and dry air descend to, and detrain at, their LNB. In a similar vein we present a more general version of the intrusion model in section 3g, allowing for a range of LNBs associated with different mixtures of intrusion and crater air.
Because of the very low moisture content of the air at the Meteor Crater, the effect of humidity on buoyancy can be neglected. During the night of 22–23 October 2006 (the temperature evolution of which is shown in Fig. 1), specific humidity values outside and in the crater were within the range of 2.2–2.7 g kg−1. Specific humidity differences between inflowing and crater air did not exceed 0.2 g kg−1, giving a contribution to buoyancy of less than 0.04 K.
b. Analytical solution of intrusion cooling
The solution obtained so far, as given by (13) and (15), represents intrusion cooling only and thus cannot be expected to closely match observations in the real atmosphere, where in situ cooling takes place as well. However, during the night of 26–27 October 2006 an unusually late cold-air intrusion was observed in the Meteor Crater, after the stratification had already reached dθ/dz ≈ 0.04 K m−1 (Whiteman et al. 2010, their Fig. 6). We use this value as an initial stratification. Based on observations of the magnitude of temperature drop at rim level on the upwind side of the crater, as indicated in the Whiteman et al. (2010) figure, we set the initial temperature deficit of the inflowing air to 5 K. The other model parameters are set as before, that is, L = 1200 m, H − h = 170 m, and DIN = 10 m.
Figure 3 shows that the crater atmosphere reacts quickly to the cold-air intrusion. The evolution of the temperature profile assumes different forms depending on the strength of detrainment. (Note that in all figures temperature rather than potential temperature is shown, to make identification of isothermalcy easier.) If CD is small, most of the cooling is due to lifting, which creates a sequence of almost linear profiles above an adiabatic layer growing from the LNB upward. This mode of cooling is similar to a filling-up process, like liquid poured into a container (Fig. 3a). If CD gets bigger, more of the cooling is taking place at upper levels, creating curved temperature profiles (Figs. 3b,c). Mathematically, the curvature can become so strong that a superadiabatic stratification develops. There is no physical mechanism in the above equations that would inhibit that. In the real atmosphere, buoyancy-driven mixing would quickly remove such unstable gradients. The simplest way to model this would be by using dry-adiabatic adjustment. This method has been used in the numerical model described below. However, a closed analytical solution is then no longer possible.
It can be seen from Fig. 3 that the bulk cooling after a given time differs in the three cases shown. The strength of detrainment not only determines the partitioning between the detrainment and lifting contributions, but also influences the total amount of cooling. This is because the net cooling depends not just on the temperature of the inflowing air but on the temperature difference between inflowing and outflowing air [(2)]. For a given temperature of the inflowing air (in contrast to a given temperature deficit), the cooling is stronger if the air at upper levels in the crater remains warmer. One could imagine a (hypothetical) extreme case with a very large value of CD, where all the cooling takes place in the uppermost layer. The temperature of this layer would drop rapidly to a value close to the temperature of the inflowing air, and little further cooling would occur. In reality, vertical mixing removes unstable gradients, so that the lowest possible temperature in the uppermost layer, and thus the smallest intrusion cooling, would be one corresponding to a dry-adiabatic stratification.
3. Numerical model results
a. Parameterization of in situ cooling
The cooling due to the cold-air inflow, if given sufficient time, evidently leads to a dry-adiabatic temperature profile. In the real atmosphere there is the additional surface-driven cooling, which tends to increase stability. We now investigate under which conditions the combined effect of both processes results in a near-isothermal temperature profile.
Prescribing the in situ cooling enables us to focus on the phenomenon of the near-isothermal stratification but also limits the generality of some of our results. A unified treatment of the downslope flow driven by in situ cooling and the intruding flow would be more satisfying. However, one would also have to model the further cooling of air once it arrives at the crater floor and becomes virtually stagnant. The nonlocal nature of turbulence generation under strongly stratified conditions poses a challenge for mesoscale models that rely on Monin–Obukhov similarity theory (Pinto et al. 2006). Large-eddy simulations (LES) have been used with some success in the modeling of daytime thermally driven flows on small scales in complex terrain (Chow et al. 2006; Weigel et al. 2006). However, even in an LES model, nighttime downslope flows are difficult to represent because of their smaller depth, intermittent turbulence, and the generally smaller scale of energy-containing eddies in stable boundary layers.
b. Combined intrusion and in situ cooling
The combined intrusion and in situ cooling is modeled on the basis of (7), like in the case of the intrusion-only model, but solved numerically. The discretization consists of 100 layers in the vertical (Δz = 1.7 m), and the heat budget equation in (7) is used in flux form to ensure conservation of heat. To make the simulation results comparable to the observations on 22–23 October 2006 (Fig. 1), we start with a vertically uniform potential temperature corresponding to a temperature of 16°C at the basin floor. The intrusion cooling is not “switched on” immediately, but after 2 h of undisturbed in situ cooling. Thus after 2 h the profile is still the same in all cases. The outside air at rim level is assumed to have an initial temperature deficit at t = 0 of 1 K. Its temperature then decreases linearly over time at a rate of 1 K h−1. This rate is based on the observed decrease of temperature at the rim on 22–23 October 2006. All other model parameters are kept at the values used above.
Figure 4 shows the evolution of the temperature profile for the combined in situ and intrusion cooling for increasing values of inflow speed. It can be seen that a quasi-linear, near-isothermal layer develops above the stable surface layer. The complete formation of this quasi-linear layer takes a little more than 3 h in the case of stronger inflow (2 m s−1). For weaker inflow (1 m s−1) the quasi-linear layer develops as well, but after a proportionally longer time. Comparing Figs. 4b and 4c, it can be seen that the profile at 2 h after switch-on in the weak inflow case is similar to that at 1 h after switch-on in the stronger inflow case.
c. Extension of simplified topography to three dimensions
d. Combined intrusion and in situ cooling for realistic topography
Another step toward realism is to replace the vertical sidewalls of the cylindrical basin by the actual area–height distribution of the Meteor Crater, derived from a high-resolution digital elevation dataset. The two main effects of this modification are a reduction of total air volume in the Meteor Crater compared to the cylindrical basin by ~40%, and the fact that, due to the reduced cross-sectional area, a given vertical mass flux is equivalent to a higher vertical velocity at lower levels where the basin is less wide. Figure 6 shows the resulting temperature evolution for an inflow speed of 2 m s−1. As in the vertical sidewall case, a quasi-linear profile develops. The speed of formation is increased in accordance with the reduced volume of the crater compared to the cylindrical basin. The quasi-linear stratification, which is completely established after 2 h of inflow, is now, however, closer to dry-adiabatic than to isothermal.
e. Physical significance of the near-isothermal stratification
In order for the above line of reasoning to be valid, most of the intrusion cooling must take place through an adiabatic rising motion rather than direct detrainment. The numerical model results indicate that this is the case for values of the detrainment coefficient of ~0.05 or smaller. For larger detrainment coefficients, an adiabatic layer develops because the cooling at upper levels in the crater exceeds that at lower levels and leads to buoyancy-driven vertical mixing. The laboratory results of Baines (1999) on downslope flows into a stratified environment show that detrainment coefficients are typically <0.04, which would lend some support to this picture. However, the slopes in Baines’ laboratory experiments are smooth and uniform, unlike the inhomogeneous and partially rugged surface of the Meteor Crater, which makes direct quantitative comparison difficult.
f. Azimuthal dependencies
In the real atmosphere, the air that enters the crater on the upwind side will be colder than the air at some distance along the perimeter. This is because the cross-rim flow component is largest on the upwind side; hence, the flow is able to lift air from relatively lower (potentially colder) levels up and into the crater. When air with a whole spectrum of temperature deficits enters the crater, the level of neutral buoyancy becomes an entire range of levels. Near the sides of the crater, the inflowing air will have only a small temperature deficit and will have already detrained at higher levels in the crater, whereas air at the upwind side will penetrate deeper and detrain closer to the crater floor.
We can use the mass flux model to assess what effect this buoyancy sorting has on the temperature evolution in the crater. Lacking detailed information about the along-rim dependence of temperature deficit of the inflowing air, we make the assumption that it follows a cosine law. Note that this is a cosine dependence in addition to the one of the cross-rim flow component, which has already been accounted for implicitly, leading to the factor 4/π in (19).
Figure 7 illustrates the resulting temperature evolution. Comparison with Fig. 6 shows the expected reduction in the degree of cooling, since only the air impinging perpendicularly onto the crater rim now has the maximum temperature deficit. Also, the characteristic filling-up signature of the intrusion cooling, which is evident in Figs. 4 and 6, is less pronounced. The temperature profile develops a stratification that is closer to isothermalcy than in the case without azimuthal variations. During the evolution toward the near-isothermal state, the stratification remains more uniform in the vertical because of the more distributed detrainment associated with buoyancy sorting.
g. Sinking of detrained air
Up to now it has been assumed that the detrained inflow air is mixed completely and instantaneously with the ambient air in the crater at the level of detrainment. In the real atmosphere, however, the mixing process takes a finite time, and as long as a detrained volume of air is still cooler than the ambient air, it will continue to sink. This is what has been observed in the laboratory experiments of Baines (1999, 2001, 2005). He describes three regions: (i) the actual downslope flow layer, (ii) a mixing region containing detrained fluid, and (iii) the undisturbed ambient fluid. The conceptual model depicted in Fig. 2 takes i and iii into account, but not ii. We now ask whether the presence of this additional region substantially alters the temperature evolution in the crater.
The net effect of taking the mixing region ii into account will be a stronger downward mass flux in the crater because parcels retain some of their negative buoyancy even after detrainment. This can be built into the mass flux model by adding to the downslope mass flux M(z) as given by (5) an additional downward mass flux N(z). It is important to note that the additional mass flux cannot be prescribed independently but follows from the detrainment profile and from an assumption about the degree of mixing of the detrained air. We denote the degree of mixing by the parameter 0 ≤ ε ≤ 1, which means that a mixture will consist of ε parts intrusion air and 1 − ε parts crater air. If ε = 0, there is complete instantaneous mixing, which leads back to the previous model. If ε = 1, there is no mixing, which means the detrained air sinks to the same level of neutral buoyancy as the air that has remained in the downslope flow. This is equivalent to having no detrainment at all, and hence also leads back to the previous model, for the special case CD = 0.
What are realistic values for the mixing parameter ε? Visualization of the mixing region in the tank experiments of Baines (2005) shows that its depth is typically 5–10 times greater than the depth of the undiluted katabatic flow. Based on this we may assume that typical values for ε are in the range of 0.1–0.2. Figure 8 shows the temperature evolution for ε = 0.2 (dashed lines) compared to the reference case ε = 0.0 (continuous lines) that was already shown in Fig. 3c. The changes are rather subtle. It can be seen that the cooling is somewhat reduced in the uppermost layers, and slightly increased at heights between 100 and 150 m. The profiles are more linear, and the total amount of cooling is larger. This is because the temperature at the top remains a bit higher over the first several hours, keeping the temperature difference between inflowing and outflowing air somewhat higher.
4. Summary and conclusions
A mass flux model has been developed to explain the unexpected cooling behavior observed in Arizona’s Meteor Crater. This cooling behavior, which differs from that found in other small basins, is characterized by the formation of a deep, near-isothermal layer above a shallow, stable surface layer. The conceptual model of cold-air intrusions proposed by Whiteman et al. (2010) is confirmed by the results of the mass flux model presented in this paper. The model also offers an explanation of the near-isothermal stratification in terms of increasingly colder intruding air and the finite time spent by air parcels in the compensating adiabatic rising motion. It has been shown that the detrainment parameter largely determines the characteristics of the cooling process, which may be described in terms of filling-up and “destabilization” modes. Through its effect on the shape of the temperature profile, the detrainment in a subtle way even affects the total cooling of the crater. An extended version of the mass flux model, which takes into account the sinking of detrained air, gives only slightly different results.
While the mass flux model has proven to be a useful tool for gaining a better understanding of the processes involved, its limitations in simulating the real atmosphere must be addressed. The main simplification is the treatment of in situ cooling, which has been prescribed based on observations and on conditions usually seen in other small closed basins. One possible approach would be to model the in situ cooling as part of the same mass flux framework since it leads to locally driven downslope flows. This could make the mass flux framework presented here useful for a wide range of applications (e.g., in the mapping of cold-air pooling potential). Another simplification concerns the fact that the model is essentially based on conservation of mass and heat but does not include the momentum budget. Hence, dynamic effects, such as the undershooting of intrusion air or the formation of gravity waves, are not taken into account. Nevertheless, the correspondence between model results and observations suggests that the most important processes have been captured.
Our modeling elucidates the physical processes that lead to changes in the initial temperature structure within a basin caused by cold-air intrusions over the basin rim. It should be noted that cold-air intrusions occur with other atmospheric phenomena in mountain areas (frontal passages, cold-air sources from adjacent watersheds, cloud waterfalls that form over ridges and passes, etc.). It is hoped that these initial simulations will provide an impetus to investigate other instances of the cold-air intrusion phenomenon.
Acknowledgments
We thank the Barringer Crater Company and Meteor Crater Enterprises, Inc., for access to the field site and the U.S. National Science Foundation Physical and Dynamic Meteorology Division for Research Grants ATM-0444205, ATM-0521776, and ATM-0837870. The National Center for Atmospheric Research’s Earth Observing Laboratory provided equipment, field support, and quality-controlled datasets. We particularly thank Maura Hahnenberger (University of Utah) and the staff of the NCAR Integrated Soundings System and Integrated Surface Flux System for their field and data processing assistance and Maura Hahnenberger for scientific discussions. One of the authors (ML) was supported by a Doktoratsstipendium from the Nachwuchsförderung der Universität Innsbruck and a Stipendium für kurzfristige wissenschaftliche Arbeiten im Ausland (Universität Innsbruck). Another (SWH) received support from Army Research Office Grant 52734-EV.
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